Consistency of Cheeger and Ratio Graph Cuts

TL;DR

This paper proves that clustering algorithms based on Cheeger and ratio graph cuts are consistent, meaning they accurately recover true data partitions as sample size grows, under specific scaling conditions.

Contribution

It establishes the theoretical consistency of Cheeger and ratio cut-based clustering methods and specifies optimal scaling conditions for their convergence.

Findings

Minimizers of graph cuts converge to continuum cuts as sample size increases

Sharp scaling conditions for connectivity radius are identified

Numerical experiments validate theoretical results and optimality in 2D

Abstract

This paper establishes the consistency of a family of graph-cut-based algorithms for clustering of data clouds. We consider point clouds obtained as samples of a ground-truth measure. We investigate approaches to clustering based on minimizing objective functionals defined on proximity graphs of the given sample. Our focus is on functionals based on graph cuts like the Cheeger and ratio cuts. We show that minimizers of the these cuts converge as the sample size increases to a minimizer of a corresponding continuum cut (which partitions the ground truth measure). Moreover, we obtain sharp conditions on how the connectivity radius can be scaled with respect to the number of sample points for the consistency to hold. We provide results for two-way and for multiway cuts. Furthermore we provide numerical experiments that illustrate the results and explore the optimality of scaling in dimension two.